While there are obvious dangers to the stability of the region if the Saudis decide to obtain nuclear weapons as well, Totten here notes:

There’s another interesting angle here, though, that hardly anyone seems to notice. Israel is supposedly the mortal enemy of the Arabs. Right? So how come no Arab state bothered getting nuclear weapons after Israel acquired the bomb? Either the Arab war against Israel is less serious than the conventional wisdom would have it, the Arab-Persian conflict is more serious than the conventional wisdom would have it, or both. [emphasis added]